Chapter 10: Q. 434 (page 1065)

In the following exercises, solve for x.
$\log (x - 1) - \log (3 x + 5) = - \log x$

Short Answer

Expert verified

The value of $x$ for the equation role="math" localid="1645038702854" $\log (x - 1) - \log (3 x + 5) = - \log x$ is $5$

Step by step solution

Step 1. Given data

The given Logarithmic Equation is $\log (x - 1) - \log (3 x + 5) = - \log x$

Step 2. Apply the quotient properties of the logarithm

Use quotient properties of the logarithm on the left side and power property on the right side

$\log (x - 1) - \log (3 x + 5) = - \log x \log \frac{(x - 1)}{(3 x + 5)} = \log x^{- 1} \log \frac{(x - 1)}{(3 x + 5)} = \log \frac{1}{x}$

Step 3. Apply one to one property

Use one to one property of logarithms, if $\log_{a} (m) = \log_{a} (n)$

then $m = n$

so $\log \frac{(x - 1)}{(3 x + 5)} = \log \frac{1}{x} \frac{(x - 1)}{(3 x + 5)} = \frac{1}{x} x (x - 1) = (3 x + 5) x^{2} - x = 3 x + 5 x^{2} - 4 x - 5 = 0 (x + 1) (x - 5) = 0$

so $x = 5, - 1$

the logarithm of a negative number is not possible

so $x = 5$

Step 4. verification

Substitute $5$ for x in the equation

$\log (x - 1) - \log (3 x + 5) = - \log x \log (5 - 1) - \log (3 (5) + 5) \overset{?}{=} - \log 5 \log 4 - \log (15 + 5) \overset{?}{=} \log 5^{- 1} \log 4 - \log 20 \overset{?}{=} \log \frac{1}{5} \log (\frac{4}{20}) \overset{?}{=} \log \frac{1}{5} \log \frac{1}{5} = \log \frac{1}{5}$

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In the following exercises, solve for x.
$\log (x - 1) - \log (3 x + 5) = - \log x$

Short Answer

Step by step solution

Step 1. Given data

Step 2. Apply the quotient properties of the logarithm

Step 3. Apply one to one property

Step 4. verification

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In the following exercises, solve for x.logx-1-log3x+5=-logx

Short Answer

Step by step solution

Step 1. Given data

Step 2. Apply the quotient properties of the logarithm

Step 3. Apply one to one property

Step 4. verification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Applied Mathematics

Geometry

Calculus

Pure Maths

Study anywhere. Anytime. Across all devices.

In the following exercises, solve for x.
$\log (x - 1) - \log (3 x + 5) = - \log x$